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Projection of a = alpha i + 3j - k on c is 10/3 and b x c = -6i + 10j + 7k: find alpha and beta

The same question, asked 3 times: 4 May 2023 S1*, 9 May 2023 S2*, 26 Apr 2025 S1. Every wording below, oldest first.

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  1. Question 1

    4 May 2023 S1

    112
    Let a⃗=αi^+3j^−k^\vec{a}=\alpha\hat{i}+3\hat{j}-\hat{k}, b⃗=3i^−βj^+4k^\vec{b}=3\hat{i}-\beta\hat{j}+4\hat{k} and c⃗=i^+2j^−2k^\vec{c}=\hat{i}+2\hat{j}-2\hat{k}, where α,β∈R\alpha,\beta\in\mathbb{R}, be three vectors. If the projection of a⃗\vec{a} on c⃗\vec{c} is 103\frac{10}{3} and b⃗×c⃗=−6i^+10j^+7k^\vec{b}\times\vec{c}=-6\hat{i}+10\hat{j}+7\hat{k}, then the value of 2α+β2\alpha+\beta is

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  2. Question 2

    9 May 2023 S2

    105
    Let a⃗=αi^+3j^−k^\vec{a} = \alpha\hat{i} + 3\hat{j} - \hat{k}, b⃗=3i^−βj^+4k^\vec{b} = 3\hat{i} - \beta\hat{j} + 4\hat{k} and c⃗=i^+2j^−2k^\vec{c} = \hat{i} + 2\hat{j} - 2\hat{k}, where α,β∈R\alpha, \beta \in \mathbb{R}, be three vectors. If the projection of a⃗\vec{a} on c⃗\vec{c} is 103\frac{10}{3} and b⃗×c⃗=−6i^+10j^+7k^\vec{b} \times \vec{c} = -6\hat{i} + 10\hat{j} + 7\hat{k}, then the value of α2+β2−αβ\alpha^2 + \beta^2 - \alpha\beta is equal to

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  3. Question 3

    26 Apr 2025 S1

    110
    Let aˉ=αi^+3j^−k^,bˉ=3i^−j^+βk^\bar{a}=\alpha\widehat{i}+ 3\widehat{j}-\widehat{k},\bar{b}= 3\widehat{i}-\widehat{j}+\beta\widehat{k} and c‾=i^+2j^−2k^\overline{c}=\widehat{i}+ 2\widehat{j}- 2\widehat{k} where α,β∈R\alpha,\beta\in\mathbb{R}, be three vectors. If the projection of aˉ\bar{a} on cˉ\bar{c} is 103\frac{10}{3} and b‾×c‾=−6i^+10j^+7k^\overline{b}\times\overline{c}= - 6\widehat{i}+ 10\widehat{j}+ 7\widehat{k}, then the value of (α+β)(\alpha+\beta) is equal to

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